Question

A photon of violet light has a wavelength of \(423 \mathrm{nm}\). Calculate (a) the frequency. (b) the energy in joules per photon. (c) the energy in kiloioules per mole.

Short answer

Question: A photon has a wavelength of 423 nm. Find (a) its frequency, (b) its energy in joules per photon, and (c) its energy in kilojoules per mole. Answer: (a) The frequency of the photon is \(7.09 \times 10^{14} \mathrm{s}^{-1}\). (b) The energy per photon is \(4.70 \times 10^{-19} \mathrm{J}\). (c) The energy per mole is \(283 \mathrm{kJ/mol}\).

Step-by-step solution

Calculate the frequency

First, we need to calculate the frequency of the photon. The wave equation is given by c = λν. Rearranging the formula to solve for the frequency, we get ν = c / λ. The speed of light (c) is \(3.0 \times 10^8 \mathrm{m/s}\), and the wavelength (λ) is \(423 \mathrm{nm}\) or \(423 \times 10^{-9} \mathrm{m}\). Now, we can substitute the values and find the frequency: ν = \((3.0 \times 10^8 \mathrm{m/s}) / (423 \times 10^{-9} \mathrm{m})\) = \(7.09 \times 10^{14} \mathrm{s}^{-1}\).

Calculate the energy per photon

Now, we'll use the energy equation, E = hν, to calculate the energy per photon. Planck's constant (h) is \(6.63 \times 10^{-34} \mathrm{J s}\). Using the calculated frequency from Step 1, we can find the energy: E = \((6.63 \times 10^{-34} \mathrm{J s}) \times (7.09 \times 10^{14} \mathrm{s}^{-1})\) = \(4.70 \times 10^{-19} \mathrm{J}\).

Calculate energy per mole

Lastly, we need to convert the energy per photon to energy per mole. To do this, we'll use Avogadro's number (N_A), which is \(6.022 \times 10^{23} \mathrm{mol}^{-1}\). We can multiply the energy per photon by Avogadro's number to find the energy per mole: E_mole = \((4.70 \times 10^{-19} \mathrm{J}) \times (6.022 \times 10^{23} \mathrm{mol}^{-1})\) = \(283 \mathrm{kJ/mol}\). So, the final answers are: (a) Frequency: \(7.09 \times 10^{14} \mathrm{s}^{-1}\) (b) Energy per photon: \(4.70 \times 10^{-19} \mathrm{J}\) (c) Energy per mole: \(283 \mathrm{kJ/mol}\)